I stumbled upon this video by Vincent Floderer demonstrating how to fold his mushroom. While you might argue whether it's still origami or not, it is definitely an interesting technique.
I'll avoid the discussion by simply calling it "crumpling". Anyone know what crumple means in Japanese? Google translate suggest: 揉む (crumpling) or 揉む紙 (crumpling paper).

This mushroom looks great...I will be trying it out shortly....I hope mine comes out as good as yours. By the way, I thought I'd send you the link of another video (Vincent Floderer's design, again) on the 'crumpling technique' This one of of Vincent Floderer's sea urchin (I think!) and is folded or crumpled rather from a napkin!

Try baking paper instead. Paper napkins also work quite well, but you have to be extremely careful not to add too much water when forming the stem, and you probably need to stabilize it until it's fully dried.